Question: Simplify the following expression and state the condition under which the simplification is valid. $t = \dfrac{-10a^2 - 10a + 300}{-8a^2 + 32a + 480}$
Answer: First factor out the greatest common factors in the numerator and in the denominator. $ t = \dfrac {-10(a^2 + a - 30)} {-8(a^2 - 4a - 60)} $ $ t = \dfrac{10}{8} \cdot \dfrac{a^2 + a - 30}{a^2 - 4a - 60} $ Simplify: $ t = \dfrac{5}{4} \cdot \dfrac{a^2 + a - 30}{a^2 - 4a - 60}$ Next factor the numerator and denominator. $ t = \dfrac{5}{4} \cdot \dfrac{(a + 6)(a - 5)}{(a + 6)(a - 10)}$ Assuming $a \neq -6$ , we can cancel the $a + 6$ $ t = \dfrac{5}{4} \cdot \dfrac{a - 5}{a - 10}$ Therefore: $ t = \dfrac{ 5(a - 5)}{ 4(a - 10)}$, $a \neq -6$